An implicit algorithm for the dynamic study of nonlinear vibration of spur gear system with backlash

Abstract

In this work, we propose some regularization techniques to adapt the implicit high order algorithm based on the coupling of the asymptotic numerical methods (ANM) (Cochelin et al., Méthode Asymptotique Numérique, Hermès-Lavoisier, Paris, 2007; Mottaqui et al., Comput. Methods Appl. Mech. Eng. 199 (2010) 1701–1709; Mottaqui et al., Math. Model. Nat. Phenom. 5 (2010) 16–22) and the implicit Newmark scheme for solving the non-linear problem of dynamic model of a two-stage spur gear system with backlash. The regularization technique is used to overcome the numerical difficulties of singularities existing in the considered problem as in the contact problems (Abichou et al., Comput. Methods Appl. Mech. Eng. 191 (2002) 5795–5810; Aggoune et al., J. Comput. Appl. Math. 168 (2004) 1–9). This algorithm combines a time discretization technique, a homotopy method, Taylor series expansions technique and a continuation method. The performance and effectiveness of this algorithm will be illustrated on two examples of one-stage and two-stage gears with spur teeth. The obtained results are compared with those obtained by the Newton–Raphson method coupled with the implicit Newmark scheme.